Algebra
How To Calculate Exponential Growth
Exponential growth happens when a quantity is multiplied by the same factor every period. Interest, population and radioactive decay all follow the same shape once the numbers are set up.
Quick Answer
A = P x (1 + r)^t, where r can be negative for decay
- A
- The final value
- P
- The starting value
- r
- The rate per period, negative for decay
- t
- The number of periods
Raise one plus the rate to the power of the number of periods, then multiply by the starting value. A thousand growing at five percent per period becomes 1628.89462678 after ten periods, and the doubling time is 14.20669908 periods.
What Is Exponential Growth?
Exponential growth is repeated multiplication. Linear growth adds the same amount each period, while exponential growth multiplies by the same factor, and the difference compounds quietly until it becomes enormous.
The formula A = P (1 + r)^t is the whole subject. The base is one plus the rate, the exponent is the number of periods, and the multiplication by P sets the scale.
A negative rate gives exponential decay. The same formula then shrinks the value, and a rate of minus one hundred percent drives it to zero in a single step.
The doubling time is the number of periods the value needs to double. It is the natural logarithm based ratio log of two over the log of the factor, and it is constant no matter where you start.
The rule of 72 is a shortcut for the same thing. Divide seventy two by the percentage rate and you get an approximate doubling time that is good enough for mental arithmetic.
Compound interest is exponential growth with a familiar name. Interest earned in one period earns interest in the next, which is why the balance curves upward rather than rising in a straight line.
Population growth follows the same shape while resources allow. Once food or space runs out the rate falls and the curve straightens into the logistic shape, but the early part is exponential.
Radioactive decay is the mirror image. The fraction surviving falls by the same ratio every half life, which is why half life is constant and the number of atoms does not predict when the last one decays.
Continuous compounding replaces the discrete factor with the base of natural logarithms raised to the rate times time. For the same nominal rate the continuous figure is always slightly higher.
Extrapolating far beyond the data is the classic trap. A small rate compounded over many periods produces huge numbers, but only while the underlying assumption of a constant rate survives.
The units of the rate must match the units of the period. A rate quoted per month cannot be applied to a number of years without converting first, and mixing the two is the most common error in financial modelling.
The base of natural logarithms appears when compounding becomes continuous. Instead of multiplying by a factor once per period, the factor is applied infinitely often within each period, and the limit is that base raised to the rate times the time.
Exponential growth looks flat before it looks vertical. A curve built on 1.05 to the power of ten is almost indistinguishable from a straight line across the first few periods, which is why exponential change is so consistently under-anticipated.
Half of the total growth in any doubling window happens in the last period of that window. The second half of a doubling takes as long as everything that came before it, and that is the counter-intuitive part.
Formula
A = P x (1 + r)^t
Formula page →The value after t periods of multiplying by the same factor.
| Symbol | Meaning | Unit | Notes |
|---|---|---|---|
| A | Final value | units | The value after the periods have elapsed. |
| P | Starting value | units | The value at the beginning. |
| r | Rate | unitless | The rate per period, expressed as a fraction. |
| t | Periods | periods | The number of periods elapsed. |
t2 = log(2) / log(1 + r)
Formula page →How many periods the value needs to double.
| Symbol | Meaning | Unit | Notes |
|---|---|---|---|
| t2 | Doubling time | periods | The number of periods to double the value. |
| r | Rate | unitless | The rate per period, expressed as a fraction. |
How To Calculate Exponential Growth
- 1
Write the rate as a fraction
Five percent is 0.05. The formula needs the fraction, not the percentage, so divide by one hundred first.
- 2
Add one to the rate
One plus 0.05 is 1.05. This is the factor the value is multiplied by each period, and it is the base of the exponent.
- 3
Raise the factor to the number of periods
1.05 to the power of ten is 1.62889463. Raising to a power is repeated multiplication, which is exactly what compounding means.
- 4
Multiply by the starting value
One thousand times 1.62889463 is 1628.89462678, so the value has grown by 628.89462678 over the ten periods.
- 5
Check the doubling time
Log of two divided by log of 1.05 is 14.20669908 periods, which the rule of 72 approximates as 72 over 5, or about 14.4.
Examples
Example 1: One thousand growing at five percent for ten periods
- Starting value
- 1000
- Rate
- 5%
- Periods
- 10
| Step | Calculation | Result |
|---|---|---|
| Rate as a fraction | 5 / 100 | 0.05 |
| Growth factor | 1 + 0.05 | 1.05 |
| Factor raised to ten periods | 1.05^10 | 1.62889463 |
| Final value | 1000 x 1.62889463 | 1628.89462678 |
| Total growth | 1628.89462678 - 1000 | 628.89462678 |
| Doubling time in periods | log(2) / log(1.05) | 14.20669908 |
Result: Ten periods at five percent turns 1000 into 1628.89462678, a gain of 628.89462678. The doubling time is 14.20669908 periods, so the value more than doubles by period 15.
Example 2: A sample decaying at twelve percent for six periods
- Starting value
- 500
- Rate
- -12%
- Periods
- 6
| Step | Calculation | Result |
|---|---|---|
| Rate as a fraction | -12 / 100 | -0.12 |
| Decay factor | 1 + -0.12 | 0.88 |
| Factor raised to six periods | 0.88^6 | 0.46440409 |
| Final value | 500 x 0.46440409 | 232.20204339 |
| Total change | 232.20204339 - 500 | -267.79795661 |
| Doubling figure for the decay factor | log(2) / log(0.88) | -5.42227098 |
Result: A twelve percent rate applied as a loss leaves 232.20204339 of the original 500 after six periods, a fall of 267.79795661. The doubling calculation returns -5.42227098 because the factor is below one; its magnitude, 5.42227098, is the half life in periods.
Example 3: A town of five thousand growing at two point one percent for 25 years
- Starting value
- 5000
- Rate
- 2.1%
- Periods
- 25
| Step | Calculation | Result |
|---|---|---|
| Rate as a fraction | 2.1 / 100 | 0.021 |
| Growth factor | 1 + 0.021 | 1.021 |
| Factor raised to 25 periods | 1.021^25 | 1.68129357 |
| Final value | 5000 x 1.68129357 | 8406.46785243 |
| Total growth | 8406.46785243 - 5000 | 3406.46785243 |
| Doubling time in years | log(2) / log(1.021) | 33.35238175 |
Result: A town of 5000 growing at 2.1 percent a year reaches 8406.46785243 after 25 years, an increase of 3406.46785243. The doubling time is 33.35238175 years, so the population doubles roughly once in a generation.
Calculator
Final value
1,628.8946
- Total growth
- 628.8946
- Growth multiple
- 1.6289
- Doubling time in periods
- 14.2067
Values update as you type. This calculator covers the single scenario its formula assumes — see Common Mistakes for what it leaves out.
Prefer a full-width tool? Open the Exponential Growth calculator page.
Common Mistakes
Using the percentage instead of the fraction
The formula expects 0.05, not 5. Entering 5 as the rate makes the factor 6 instead of 1.05, and six to the power of ten is about 60 million against an exact 1.63, so the answer is out by a factor of roughly 37 million.
Forgetting to add one to the rate
The base is one plus the rate. Using 0.05 as the base shrinks the value toward zero instead of growing it, because anything below one raised to a power falls.
Applying the rate once for the whole time
Raising the factor to the power of the periods IS the compounding. A rate applied once is simple interest, not exponential growth.
Assuming a doubling time that shortens
The doubling time is constant for a constant rate. It does not halve as the value grows, which is precisely why the curve accelerates.
Confusing periods with years
A monthly rate needs the number of months, not years. Two years of monthly compounding is 24 periods, and using 2 changes the answer by a factor of thousands.
Extrapolating past the point of validity
Constant rates rarely survive for long. A population cannot double indefinitely, so projecting fifty doubling times is arithmetic without a model.
FAQ
What is the formula for exponential growth?
It is A = P (1 + r)^t, where P is the starting value, r is the rate per period and t is the number of periods. A negative r gives decay.
How do I calculate a doubling time?
Divide the logarithm of two by the logarithm of the growth factor. At five percent per period the doubling time is about 14.21 periods.
What is the rule of 72?
Dividing seventy two by the percentage rate gives a quick estimate of the doubling time. At five percent it gives about 14.4, against an exact 14.21.
What is the difference between exponential and linear growth?
Linear growth adds a fixed amount each period while exponential growth multiplies by a fixed factor. The exponential curve passes any linear one eventually.
Can exponential growth go on forever?
Not in practice. Resources, competition or saturation slow the rate, and the curve flattens into the logistic shape.
How is this different from compound interest?
It is the same formula. Compound interest just names the starting value the principal, the rate the annual rate and the periods the years.
References
- [1]Wikipedia, Exponential growth — https://en.wikipedia.org/wiki/Exponential_growth
- [2]Wikipedia, Doubling time — https://en.wikipedia.org/wiki/Doubling_time
- [3]Wikipedia, Compound interest — https://en.wikipedia.org/wiki/Compound_interest